What a square root is and why you need it
A square root is the number that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3, because 3 × 3 = 9. The square root of 16 is 4, because 4 × 4 = 16. You write it with the symbol √, so √9 = 3.
You need square roots when you're working with areas, distances, or any measurement where something was multiplied by itself. If you know a square has an area of 25 square feet, the square root tells you each side is 5 feet long. In cooking, square roots help scale recipes that depend on surface area. In construction and home repair, they're essential for calculating diagonal distances and checking that corners are truly square.
You can find square roots three ways: recognizing perfect squares (numbers that have whole-number roots), using a calculator, or computing them by hand using long division. Most people use a calculator now, but understanding how square roots work makes you faster at estimating and spotting mistakes.
Key Takeaways
- Perfect squares like 4, 9, 16, 25, 36, 49, 64, 81, and 100 have whole-number square roots that are worth memorizing.
- A scientific or standard calculator gives you the exact square root in one keystroke using the √ button.
- The long-division method for computing square roots by hand takes practice but works for any number and helps you understand what a square root actually is.
- For non-perfect squares, you can estimate by finding the two perfect squares your number falls between, then narrow down from there.
- Checking your answer by multiplying it by itself tells you when ready whether your result is correct.
Memorizing perfect squares up to 100
The fastest way to compute a square root is to recognize it when ready. Memorize these ten perfect squares and their roots, and you'll handle most everyday situations without a calculator:
| Number | Square Root |
|---|---|
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 49 | 7 |
| 64 | 8 |
| 81 | 9 |
| 100 | 10 |
| 121 | 11 |
Spend a few minutes drilling these. Once they're automatic, you can estimate square roots of other numbers by recognizing which two perfect squares they fall between. If someone asks for the square root of 50, you know it's between √49 (which is 7) and √64 (which is 8), so the answer is between 7 and 8—closer to 7.
Using a calculator to find square roots
On a standard or scientific calculator, finding a square root takes two steps. First, enter the number you want the square root of. Then press the button labeled √ (or sometimes √x or the second-function key followed by x²). The calculator displays the answer when ready.
For example, to find the square root of 144: press 1, 4, 4, then √. The display shows 12. To find the square root of 2: press 2, then √. The display shows 1.414213562 (the exact number of decimal places depends on your calculator).
On a phone or computer, open the calculator app, switch to scientific mode if needed, enter your number, and tap the √ button. This is the method most people use in real life because it's fast and accurate. Always double-check by multiplying the answer by itself—if √144 = 12, then 12 × 12 should equal 144.
Estimating square roots without a calculator
When you don't have a calculator and the number isn't a perfect square, you can narrow down the answer by testing. This method is called trial and error, and it works well for rough estimates.
Say you need the square root of 50. You know it's between 7 and 8 because 7 × 7 = 49 and 8 × 8 = 64. Try 7.5: 7.5 × 7.5 = 56.25, which is too high. Try 7.2: 7.2 × 7.2 = 51.84, still too high. Try 7.1: 7.1 × 7.1 = 50.41, very close. So √50 ≈ 7.07 (the ≈ symbol means "approximately equal to").
This method takes longer than a calculator but teaches you how square roots behave. Each time you test a number and see whether it's too high or too low, you're narrowing the range. With practice, you can estimate to one decimal place in under a minute.
The long-division method for computing square roots by hand
The long-division method is the traditional way to compute square roots without a calculator. It's slower than trial and error but more systematic, and it works for any number. Here's how to do it step by step.
Step 1: Set up the problem. Write the number you want the square root of, and mark off pairs of digits from right to left. For example, if you're finding √529, mark it as 5|29. If you're finding √2, just write 2 with a decimal point: 2.00|00|00 (you can add as many pairs of zeros as you need for decimal places).
Step 2: Find the first digit. Look at the leftmost pair (or single digit if there's only one). Find the largest whole number whose square is less than or equal to that pair. For 5|29, the largest number whose square is ≤ 5 is 2 (because 2 × 2 = 4). Write 2 above the 5. This is your first digit of the answer.
Step 3: Subtract and bring down. Subtract 4 from 5, leaving 1. Bring down the next pair of digits (29) to make 129.
Step 4: Find the next digit. Double the digit you found in Step 2 (2 × 2 = 4). Now find a digit to append to this 4 that, when multiplied by the whole two-digit number, gives a result ≤ 129. Try 43 × 3 = 129. That works perfectly. Write 3 above the 29. This is your second digit.
Step 5: Subtract and repeat. Subtract 129 from 129, leaving 0. Since there are no more digits, you're done. The answer is 23. Check: 23 × 23 = 529. ✓
This method is tedious but reliable. Most people learn it in school and then switch to calculators. If you need to compute a square root by hand and have time, this method will get you an exact answer for perfect squares and a very accurate decimal approximation for others.
Checking your answer
No matter which method you use, always verify your result by multiplying the answer by itself. If you found that √144 = 12, multiply 12 × 12 and confirm you get 144. If you estimated √50 ≈ 7.07, multiply 7.07 × 7.07 and you should get close to 50 (you'll get 49.9849, which rounds to 50).
This check catches mistakes when ready. If your multiplication doesn't match the original number, you made an error somewhere—go back and recalculate. This habit saves time in the long run because you catch problems before they affect the rest of your work.
Common mistakes to avoid
The most common mistake is confusing square root with division by 2. The square root of 16 is 4, not 8. Remember: you're looking for the number that multiplies by itself, not the number that divides the original in half.
Another mistake is forgetting that negative numbers don't have real square roots. The square root of −9 doesn't exist in regular math because no real number times itself gives a negative result. If a problem asks for √−9, the answer is "no real solution" or "undefined."
When using a calculator, make sure you're pressing the √ button and not a different function. Some calculators have multiple square-root-related buttons, and pressing the wrong one gives you a completely different answer.
In the long-division method, the most common error is misaligning digits or forgetting to double the previous answer before finding the next digit. Write neatly and follow each step in order.
Frequently Asked Questions
What's the difference between a square root and squaring a number?
Squaring means multiplying a number by itself: 5 squared (written 5²) is 5 × 5 = 25. A square root is the reverse: √25 = 5. If squaring is the question "what do I get when I multiply 5 by itself?", the square root is the answer to "what number, when multiplied by itself, gives me 25?"
Can you have a square root of a decimal or fraction?
Yes. The square root of 0.25 is 0.5 (because 0.5 × 0.5 = 0.25). The square root of 1/4 is 1/2. The method is the same—find the number that multiplies by itself to give the original. Calculators handle decimals and fractions when ready.
Why do some square roots have decimals that never end?
Numbers like √2 and √3 are irrational, meaning their decimal representation goes on forever without repeating. √2 = 1.41421356... and never terminates. Calculators round to a certain number of decimal places, which is accurate enough for practical use. For most real-world problems, rounding to two or three decimal places is sufficient.
How do I find the square root of a really large number?
Use a calculator—that's what they're for. If you must do it by hand, the long-division method works but becomes tedious. Break the number into pairs of digits from right to left and follow the steps carefully. For numbers in the millions or larger, a calculator is the only practical choice.
What if I need a square root for a measurement in my home or garden?
Use a calculator on your phone. If you're checking whether a corner is square, you might need the Pythagorean theorem (which involves square roots), but a carpenter's square or a measuring app is faster and more reliable. For recipe scaling or any cooking math, a calculator takes the guesswork out entirely.